OFFSET
1,1
COMMENTS
59 is the second irregular prime. The corresponding entry for the first irregular prime 37 is A251782, and for the third irregular prime 67 is A299467.
The p-adic digits of the unique simple zero of the p-adic zeta function zeta_{(p,l)} with (p,l)=(59,44) were used to compute the sequence (see the Mathematica program below). This corresponds with Table A.2 in Kellner (2007). The sequence is increasing, but some consecutive entries are identical, e.g., entries 30 / 31 and 94 / 95. This is caused only by those p-adic digits that are zero.
LINKS
Bernd C. Kellner, Table of n, a(n) for n = 1..100
Bernd C. Kellner, The Bernoulli Number Page
Bernd C. Kellner, On irregular prime power divisors of the Bernoulli numbers, Math. Comp., 76 (2007), 405-441.
Wikipedia, Irregular pairs
FORMULA
Numerator(B_{a(n)}) == 0 (mod 59^n).
EXAMPLE
a(3) = 86464 because the numerator of B_86464 is divisible by 59^3 and there is no even integer less than 86464 for which this is the case.
MATHEMATICA
p = 59; l = 44; LD = {15, 25, 40, 36, 18, 11, 17, 28, 58, 9, 51, 13, 25, 41, 44, 17, 43, 35, 21, 10, 21, 38, 9, 12, 40, 43, 45, 30, 41, 0, 3, 25, 34, 49, 45, 9, 19, 48, 57, 11, 13, 29, 28, 44, 41, 37, 33, 29, 43, 8, 57, 12, 48, 15, 15, 53, 57, 16, 51, 16, 54, 30, 9, 26, 8, 49, 22, 58, 11, 42, 28, 36, 33, 45, 24, 32, 18, 12, 29, 45, 40, 27, 19, 40, 41, 11, 42, 49, 35, 41, 57, 54, 33, 0, 34, 34, 49, 6, 31}; CalcIndex[L_, p_, l_, n_] := l + (p - 1) Sum[L[[i + 1]] p^i , {i, 0, n -2}]; Table[CalcIndex[LD, p, l, n], {n, 1, Length[LD] + 1}] // TableForm
CROSSREFS
KEYWORD
nonn
AUTHOR
Bernd C. Kellner and Jonathan Sondow, Feb 10 2018
STATUS
approved